Geometry and Representation Theory Seminar

Monday, September 24th, 2018

Time: 4:30-5:30 p.m. Place: Jeffery Hall 319

Speaker: Emine Yildrim (Queen's University)

Title: The bounded derived category for cominuscule posets

Abstract: Cominuscule posets come from root posets and have connections to Lie theory and Schubert calculus. We are interested in whether the bounded derived category of the incidence algebra of a cominuscule poset is fractionally Calabi-Yau. In other words, we ask if some non-zero power of the Serre functor is a shift functor. We answer this question on the level of the Grothendieck groups. On the Grothendieck group this functor becomes an endomorphism called the Coxeter transformation. We show that Coxeter transformation has finite order for two of the three infinite families of cominuscule posets, and for the exceptional cases. Our motivation comes from a conjecture by Chapoton which states that the bounded derived category of incidence algebra of root posets is fractionally Calabi-Yau. Our result can be thought of as a parabolic analogue of Chapoton's conjecture.

Monday, September 17th, 2018

Time: 4:30-5:30 p.m. Place: Jeffery Hall 319

Speaker: Tianyuan Xu (Queen's University)

Title: Broken lines and the topological ordering of the alternating quiver of type A

Abstract: The Positivity Conjecture in cluster algebra theory states that the coefficients of the Laurent expansion of any cluster variable in a cluster algebra are always positive integers. In 2014, Gross, Hacking, Keel and Kontsevich constructed a so-called Theta function basis to prove the conjecture for all cluster algebras of geometric type. A key ingredient in the construction of the Theta functions is the broken line model. In this talk, we will discuss the broken lines associated to the alternating quiver of type A, with an emphasis on relating its combinatorial properties to the topological ordering of the quiver, the partial order obtained by taking the transitive and reflexive losure of the relation “v<w if v->w is an edge” on the vertices of the quiver.

The talk is based work in progress with Ba Nguyen, David Wehlau and Imed Zaguia.

Monday, September 10th, 2018

Time: 4:30-5:30 p.m. Place: Jeffery Hall 319

Speaker: Charles Paquette (Queen's/RMC)

Title: A quiver construction of some subalgebras of asymptotic Hecke algebras

Abstract: Lusztig defines an asymptotic Hecke algebra J from a Coxeter system (W,S). This is an algebra that is defined using the Kazhdan-Lusztig (KL) basis of the corresponding Hecke algebra of (W,S). Even though these KL bases are generally hard to understand, there is a two-sided cell C of W that gives rise to a nice subalgebra J_C of J having rich combinatorics and whose algebraic description does not use KL bases. We will see that J_C has a presentation using a quiver with relations, and this allows one to study the representation theory of J_C (and of J) from another perspective. Using quiver representations, we will see that the classification of simple modules, which falls into three categories (finite type, bounded type and unbounded type), can be characterized completely using the shape of the weighted graph G of (W,S).

Wednesday, March 28th, 2018

Time: 3:30 p.m. Place: Jeffery Hall 319

Speaker: Mike Zabrocki (York University)

Title: A Multiset Partition Algebra

Abstract: Schur-Weyl duality is a statement about the relationship between the actions of the general linear group $Gl_n$ and the symmetric group $S_k$ when these groups act on $V_n^{\otimes k}$ (here $V_n$ is an $n$ dimensional vector space). If we consider the symmetric group $S_n$ as permutation matrices embedded in $Gl_n$, then the partition algebra $P_k(n)$ (introduced by Martin in the 1990's) is the algebra which commutes with the action of $S_n$.

In this talk I will explain how an investigation of characters of the symmetric group leads us to consider analogues of the RSK algorithm involving multiset tableaux. To explain the relationship of the combinatorics to representation theory we were led to the multiset partition algebra as an analogue of the partition algebra and the dimensions of the irreducible representations are the numbers of multiset tableaux.

Monday, March 5th, 2018

Abstract: The Robinson--Schensted--Knuth correspondence is a many-faceted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of $n$ and pairs of standard Young tableaux with the same shape, which is a partition of $n$. In another (more general) version, it provides a bijection between fillings of a partition $\lambda$ by arbitrary non-negative integers and fillings of the same shape $\lambda$ by non-negative integers which weakly increase along rows and down columns (i.e., reverse plane partitions of shape $\lambda$). I will discuss an interpretation of RSK in terms of the representation theory of type $A$ quivers (i.e., directed graphs obtained by orienting a path graph). This allows us to generalize RSK to other Dynkin types (plus a choice of minuscule weight), and is related to periodicity results for piecewise-linear toggling. I will not assume familiarity with either RSK or with quiver representations. This is joint work with Al Garver and Becky Patrias.

Monday, February 5th, 2018

Time: 4:30 p.m. Place: Jeffery Hall 319

Speaker: David Wehlau (Queen's University)

Title: Khovanski Bases and Derivations

Abstract: Let $R=K[x_1,,\dots,x_m,y_1,\dots,y_m,z_1,\dots,z_m]$ be a polynomial algebera over a field $K$ of characteristic zero, Let $\Delta$ be the locally nilpotent derivation on $R$ determined by $\Delta(z_i) = y_i$, $\Delta(y_i) = x_i$ and $\Delta(x_i)=0$ for $i=1,2,\dots,m$. This is an example of

a Weitzenb\"ock derivation. We exhibit a minimal set of generators $\mathcal G$ for the algebra of

constants $R^\Delta = \ker \Delta$. We also construct a Khovanski (or sagbi) basis for this algebra.

Even though this basis is infinite our proof yields an algorithm to express any element of $R^\Delta$ as
a polynomial in the elements of $\mathcal G$. In particular, this method shows how the classical techniques of polarization and restitution may be used in combination with Khovanski bases to yield a constructive method for expressing elements of a subalgebra as a polynomials in its generators.